On the metric dimension of line graphs

TL;DR

This paper investigates the metric dimension of line graphs, providing exact formulas for certain classes like strongly connected digraphs, de Brujin, Kautz digraphs, and trees, expanding understanding of resolving sets in graph theory.

Contribution

It establishes new formulas and bounds for the metric dimension of line graphs across various graph classes, including strongly connected digraphs, de Brujin, Kautz digraphs, and trees.

Findings

(G)|-(G)| for strongly connected digraphs (except directed cycles)

Bounds (G) e (G)e for simple connected graphs with t least five vertices

Metric dimension of line graphs of trees expressed in terms of tree parameters

Abstract

Let $G$ be a (di)graph. A set $W$ of vertices in $G$ is a \emph{resolving set} of $G$ if every vertex $u$ of $G$ is uniquely determined by its vector of distances to all the vertices in $W$. The \emph{metric dimension} $\mu (G)$ of $G$ is the minimum cardinality of all the resolving sets of $G$. C\'aceres et al. \cite{Ca2} computed the metric dimension of the line graphs of complete bipartite graphs. Recently, Bailey and Cameron \cite{Ba} computed the metric dimension of the line graphs of complete graphs. In this paper we study the metric dimension of the line graph $L(G)$ of $G$. In particular, we show that $\mu(L(G))=|E(G)|-|V(G)|$ for a strongly connected digraph $G$ except for directed cycles, where $V(G)$ is the vertex set and $E(G)$ is the edge set of $G$. As a corollary, the metric dimension of de Brujin digraphs and Kautz digraphs is given. Moreover, we prove that $\lceil\log_2\Delta(G)\rceil\leq\mu(L(G))\leq |V(G)|-2$ for a simple connected graph $G$ with at least five vertices, where $\Delta(G)$ is the maximum degree of $G$. Finally, we obtain the metric dimension of the line graph of a tree in terms of its parameters.